e-small essential submodules

Let r be a commutative ring with identity, and ur be an r-module, with e = end(ur). in this work we consider a generalization of class small essential submodules namely e-small essential submodules. where the submodule q of ur is said e-small essential if q ∩w = 0 , when w is a small submodule of ur, implies that ns (w) = 0, where ns (w) = {ψ ∈ e | imψ ⊆ w}. the intersection br(u) of each submodule of ur contained in soc(ur). the br(u) is unique largest e-small essential submodule of ur, if ur is cyclic. also in this paper we study br(u) and we (u). the condition when br(u) is e-small essential, and tot ( u, u) = we (u) = j(e) are given.